Conservation of Momentum

In physics, the term conservation alludes to something which doesn’t change. This implies that the variable in a condition represents a conserved quantity. It has a similar worth both before and after the event. There are many conserved quantities in material science. They are regularly strikingly helpful for making forecasts in what might somehow or another be confounded circumstances. In mechanics, three fundamental quantities are conserved. These quantities are momentum, energy, and angular momentum.  Conservation of momentum is generally in use for depicting collisions between objects.

Introduction to Conservation of Momentum

Likewise, with the other conservation standards, there is a catch: conservation of momentum applies just to an object of an isolated system. For this situation, an isolated system is one that is not acted on by any external force to the system – i.e., there is no external impulse. Practically this means we need to incorporate the two objects and anything that applies a force to any of the objects for any timeframe in the system.

On the off chance, if i and f indicate the initial and final momentum of objects in a framework, then the principle of conservation of momentum says that

\(P_{1i}+P_{2 i}+\ldots=P_{1 f}+P_{2 f} \ldots\)

Momentum

Momentum is the result of the product of the mass and velocity of an object. It is a vector quantity, having a direction and a magnitude. If m is an object mass and v is its velocity (additionally a vector amount), at that point the momentum of an object is the result of the velocity and mass of an object.

P=mv

In SI units, momentum is estimated in kilogram meter per second.

Conservation of Momentum

It is a vector quantity. Conservation of momentum is a crucial law of physics. It expresses that the total momentum of a detached or isolated system/framework is conserved. As such, the total momentum of a system of objects stays steady during any interaction, if no external force follows up on the system. The total momentum is the vector sum of individual momenta in a system. Consequently, the component of the total momentum along any direction remains constant. Momentum stays conserved in any physical process.

Illustration of conservation of momentum in One – Dimension

Conservation of momentum can be expressed through a one-dimensional collision of two objects. Two objects of masses \(m_{1}\) and \(m_{2}\) collide with each other when they move along a straight line with velocities \(u_{1}\) and \(u_{2}\) respectively. After the collision, they acquire velocities \(v_{1}\) and \(v_{2}\) in the same direction.

Total momentum before collision, \(P_{i}=m_{1} u_{1}+m_{2} u_{2}\)

Total momentum after collision, \(P_{f}=m_{1} v_{1}+m_{2} v_{2}\)

If no other force acts on the system, the total momentum remains conserved. Therefore, \(P_{i}=P_{f}\)

\(m_{1} u_{1}+m_{2} u_{2}=m_{1} v_{1}+m_{2} v_{2}\)

Derivation of Conservation of Momentum

In the event that, if no external force is applied on the arrangement of two colliding objects. The objects apply impulse on one another for a short timespan at the point of contact. As per Newton’s third law of motion, the impulsive force applied by the first object on the second one is equal and opposite to the impulsive force applied by the second object on the first object.

Crashes are especially fascinating to dissect utilizing the conservation of momentum. This is on the grounds that collisions happen quickly. So, the time impacting objects spend connecting is short. A short association time implies that the impulse, \(F\cdot\) \(\Delta t\), because of external forces. Like, friction during the collision is very small.

During the one-dimensional collision of two objects of masses \(m_{1}\)  and \(m_{2}\), which have velocities \(u_{1}\)  and \(u_{2}\)  before collision and velocities \(v_{1}\)  and \(v_{2}\)  after the collision, the impulsive force on the first object is \(F_{21}\) (applied by the second object) and the impulsive force on the second object is \(F_{12}\) (applied by the first object). By applying Newton’s third law, the two impulsive forces are equal and

opposite. So, \(F_{21}=-F_{12}\)

If the time span of the contact is t, the impulse of the force \(F_{21}\) is equal to the change in momentum of the first object.

\(F_{21} t=m_{1} v_{1}-m_{1} u_{1}\)

The impulse of force \(F_{12}\) is equal to the momentum of the second object.

\(F_{12} t=m_{2} v_{2}-m_{2} u_{2}\)

From \(F_{21}=-F_{12}\)

\(F_{21} t=-F_{12} t\)

\(m_{1} v_{1}-m_{1} u_{1}=-\left(m_{2} v_{2}-m_{2} u_{2}\right)\)

\(m_{1} u_{1}+m_{2} u_{2}=m_{1} v_{1}+m_{2} v_{2}\)

This relation shows that momentum is conserved during the collision.

Example of Conservation of Momentum

Balloon: The small particles of gas move quickly crashing into one another and the walls of the balloon. Despite the fact that the particles themselves move quicker and slower when they lose or pick-up momentum when they collide. The total momentum of the system stays as before. Consequently, the balloon doesn’t change in size, on the off chance that we add outside energy by heating it. The balloon ought to extend in light of the fact that it builds the speed of the particles and this expands their force, thus, increasing the force applied by them on the walls of the balloon.

The recoil of a Gun: If a bullet is shot from a gun, both the bullet and the gun are at first very still i.e., the total momentum before firing is zero. The bullet gains a forward momentum when it gets discharged. As per the conservation of momentum, the gun gets a regressive momentum. The bullet of mass m is terminated with velocity v. The gun of mass M gains a retrogressive backward velocity u. Prior to terminating, the total momentum is zero. So, the total momentum after firing is also zero.

0=mv+Mv

\(u=-\frac{m}{M} v\)

Basically, u is the recoil velocity. The mass of the bullet is very less than the mass of the gun i.e., m < < M. The backward velocity of the gun is very less, u<<v.

Rocket Propulsion: Rockets have a gas chamber at one end of it. From this chamber, gas is ejected with an enormous velocity. Before the ejection of the gas, the total momentum is zero. Due to the ejection of gas from the rocket, the rocket gains a recoil velocity and acceleration in the opposite direction. This is because of the conservation of momentum.

FAQs on Conservation of Momentum

Q.1. What does the law of conservation of momentum state?

Answer: The law of conservation of momentum states that in an isolated system the total momentum of two or more bodies acting upon each other remains constant unless an external force is applied. Therefore, momentum can neither be created nor destroyed.

Q.2. Give Examples of Conservation of Momentum.

Answer: All physical processes abide by the law of conservation of momentum. Some examples are,

Collision: The collision of different objects follows the conservation of momentum and energy.

Rocket motion: The momentum of the gas particles ejected gives the rocket an opposite momentum. This is due to momentum conservation.

Ejection of a bullet from a gun: It is a consequence of conservation of momentum as if a gun experiences a recoil momentum due to the ejection of a bullet.

Q.3. The law of conservation of momentum is based on which law of motion?

Answer: The law of conservation of momentum is based on Newton’s third law of motion. This states that every force has a reciprocating equal and opposite force.

Q.4. What is the law of conservation of momentum formula?

Answer: The momentum observation principle can be mathematically represented as:
\(m_{1} u_{1}+m_{2} u_{2}=m_{1} v_{1}+m_{2} v_{2}\)
In the equation, \(m_{1}\) and \(m_{2}\) are masses of the bodies, \(u_{1}\) and \(u_{2}\)  are the initial velocities of the body. \(v_{1}\) and \(v_{2}\)  are the final velocities of the bodies.

Q.5. Does friction affect the conservation of momentum?

Answer: Yes, friction affects momentum. As friction increases, momentum decreases.